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explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach
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explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach
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Journal Article
explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach
2011
Overview
Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matérn class, provide an explicit link, for any triangulation of , between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere.
Publisher
Blackwell Publishing Ltd,Wiley-Blackwell,Blackwell,Royal Statistical Society,Oxford University Press
Subject
/ Approximate Bayesian inference
/ Exact sciences and technology
/ Gaussian
/ Gaussian Markov random fields
/ Inference from stochastic processes; time series analysis
/ Matrices
/ Modeling
/ Partial differential equations
/ Probability Theory and Statistics
/ Probability theory and stochastic processes
/ Property
/ Sannolikhetsteori och statistik
/ Sciences and techniques of general use
/ Stochastic partial differential equations
/ Studies
